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摘要: 非连续谱雷达信号是一种特殊的认知雷达信号，其频谱由多个离散的频带组成，且能够随着外界干扰的变化自适应地调整离散频带的分布结构。因此，这种信号适用于干扰密布、频谱拥堵的工作场景。非连续谱信号设计主要研究两个问题：一是如何根据干扰环境选取最优的非连续频谱结构以满足雷达抗干扰和分辨性能要求；二是如何根据最优的非连续频谱求解出时域发射信号。非连续谱雷达信号的一个典型应用是高频雷达的抗同频干扰，随着电子对抗的升级以及多电子设备共存引起的频谱拥堵问题，非连续谱信号在雷达抗干扰、电磁频谱兼容等方面日益受到重视。该文对非连续谱信号设计准则与约束、工作频带选取与塑形以及时域信号波形合成等3个方面的研究进行了归纳与总结，以促进非连续谱信号的研究与应用。Abstract: The discontinuous spectrum radar signal is a featured cognitive radar signal. Its spectrum is discontinuous and comprises multiple discrete frequency bands, and the distribution structure of the discrete frequency bands can be adapted to the change of external interference adequately. Therefore, this segmented signal is suitable for dense interference and congested spectrum based spectrum scenarios. The design of discontinuous spectrum signal is focused on two issues: (1) the optimal selection of the discontinuous spectrum structure in accordance with the interference environment to meet the requirements of radar anti-jamming and resolution performance, and (2) the solution of the time-domain emission based on the optimal discontinuous spectrum signal. A typical application of discontinuous spectrum radar signals is the anti-co-frequency interference derived of high-frequency radar. With the upgrade of electronic countermeasures and the spectrum congestion problem caused by the coexistence of multiple electronic devices, discontinuous spectrum signals are used in radar anti-jamming and electromagnetic spectrum compatibility. This paper discusses and summarizes the research on discontinuous signal design criteria and constraints, working frequency band selection and shaping, and time-domain signal waveform synthesis to promote the research and application of discontinuous spectrum signals.
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图 1 高频地波下的相位编码信号频谱
Figure 1. Phase-encoded signal spectrum under high frequency ground waves
图 2 雷达通信频谱共存复合调制信号频谱
Figure 2. Spectrum of radar communication spectrum coexistence composite modulated signal
图 7 随机矩阵的概率密度函数与其累积分布函数
Figure 7. Probability density function of random matrix and its cumulative distribution function
图 8 非连续比与峰值旁瓣以及积分旁瓣性能关系
Figure 8. Influence of discontinuous ratio on peak sidelobe/integral sidelobe performance
图 9 可用比ρ = 0.8时的双阻带峰值旁瓣(dB)
Figure 9. The PSL (dB) of double stopband under spectrum availability ratio ρ = 0.8
图 10 50组工作频带的通阻带结构以及对应的ISE与PSL性能
Figure 10. Pass-stop band structure of 50 groups of working frequency bands and corresponding ISE and PSL performance
图 11 在不同权重因子γ下IME与ISE的变化曲线
Figure 11. Variation curves of IME and ISE under different compromise factors γ
图 12 不同权重因子γ下最优功率谱密度及对应自相关函数
Figure 12. Optimal PSDs and corresponding ACFs under different trade-off factors γ
图 13 复合调制信号的功率谱密度以及自相关序列
Figure 13. Power spectral density and autocorrelation sequences of composite modulated signals
表 1 非连续谱信号典型工作场景
Table 1. Typical work scene of discontinuous spectrum signals
工作频段 场景内典型雷达 场景内典型干扰辐射源 高频频段(3～30 MHz) 高频地波雷达 固定与移动通信、广播等无线电用户、干扰机 甚高频频段(30～300 MHz)
特高频频段(300～1000 MHz)叶簇穿透雷达 电视、移动通信、广播电台 L波段(1～2 GHz) 空中交通管制雷达 频分双工LTE蜂窝通信系统、全球卫星
导航系统、5G NRS波段(2～4 GHz)
C波段 (4～8 GHz)机载预警雷达、气象雷达、战场/地面监控雷达、
船舶交通服务雷达WLAN网络、3.5 GHz TDD-LTE、5G NR 极高频频段(30～300 GHz) 民用毫米波雷达 通信基站、车辆互联网络 
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表 2 工作场景类型
Table 2. Type of work scene
场景 干扰来源 干扰时变特性 第1类 敌方干扰机等 慢时变 第2类 敌方干扰机等 快时变 第3类 协同工作的电子设备 慢时变 第4类 协同工作的电子设备 快时变
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表 3 非连续谱信号
Table 3. Typical application scenarios of discontinuous spectrum signals
分类方式 信号形式 典型信号 表达式 调制方式 相位编码 连续相位编码、离散相位编码 $ {\boldsymbol{s}} = {[s(1),s(2), \cdots ,s(N)]^{\text{T}}} = a\left( n \right){{\text{e}}^{{\text{j}}\varphi \left( n \right)}}{{\text{e}}^{{\text{j}}2\pi {f_{\text{c}}}t}} $
式中，a为信号幅度，$ \varphi $为编码相位，N为编码长度，$ {f_{\text{c}}} $为中心载频频率调制 线性调频、
非线性调频、多相编码调频${\boldsymbol{s}} = a\left( t \right){ {\text{e} }^{ {\text{j} }2\pi \left( { {f_{\text{c} } } + f\left( t \right)} \right)t} }$，式中，$ f\left( t \right) $为频率调制函数 频率编码 步进频率编码、分段步进频率编码、频率捷变编码 ${\boldsymbol{s}} = \displaystyle\sum\nolimits_{m = 0}^{M - 1} s \left( {t - m{T_r} } \right){ {\text{e} }^{ {\text{j} }2\pi {f_m}\left( {t - m{T_r} } \right)} }$，
式中，$ {f_m} $为编码频率，M为编码长度复合调制 跳频脉内线性调频、脉间跳频脉内编码、多载频相位编码 ${\boldsymbol{s}} = \displaystyle\sum\nolimits_{m = 0}^{M - 1} { {a_m}\left( t \right){ {\text{e} }^{ {\text{j} }\left( {2\pi \left( { {f_m} + {f_m}\left( t \right)} \right)t + \varphi \left( t \right)} \right)} } }$，式中，$ {f_m}\left( t \right) $为脉内频率调制函数，$ \varphi \left( t \right) $为脉内相位调制函数，$ {f_m} $为脉间编码频率 收发模式 单发单收 上述调制形式均可 – 多发多收 上述调制形式均可 ${\boldsymbol{s}} = {[{s_1},{s_2}, \cdots ,{s_l}, \cdots ,{s_L}]^{\text{T} } }$，式中，$ {s_l} $为第l 路发射信号，L为通道数 抗干扰维度 快时间域 相位编码、频率调制 – 慢时间域 频率编码、复合调制 空频联合域 单发单收、多发多收
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表 4 常见信号性能评价准则
Table 4. Common signal performance evaluation criteria
性能准则 表达式 最大化SINR 信干噪比：${\text{SINR} } = \dfrac{ {\left| {w{\boldsymbol{s}}} \right|} }{ {E\left\{ {wn} \right\} } }$(任意滤波形式)，w为雷达接收滤波器，n为干扰与噪声的和
信干噪比：${\text{SINR} } = \dfrac{ { { {\boldsymbol{s} }^{\text{H} } }{\boldsymbol{s} }{ {\boldsymbol{s} }^{\text{H} } }{\boldsymbol{s} } } }{ {E\left\{ { { {\boldsymbol{s} }^{\text{H} } }n{n^{\text{H} } }{\boldsymbol{s} } } \right\} } } = \dfrac{ { {E_{\rm{s} } } } }{ { { {\boldsymbol{s} }^{\text{H} } }{\boldsymbol{Ks} }} }$(匹配滤波形式)，$E_{\rm{s}}$为信号能量，K为干扰与噪声联合相关矩阵最小化同频干扰 同频干扰：$\left|{p}_{\varOmega }\right|\le \mu ,\varOmega \in $阻带频率位置，p为发射信号功率谱，$ \mu $为阻带阈值 分辨性能 主瓣分辨性能 积分主瓣能量：${{\rm{IME}}} = \dfrac{1}{ {|\chi (0){|^2} } }\displaystyle\int_{ - {\tau _{ {\text{main} } } }}^{ {\tau _{ {\text{main} } } }} {|\chi (\tau ){|^2}{\text{d} }\tau }$，主瓣3 dB，宽度：$2\left| {{\tau _{3{\text{ dB}}}}} \right|$，${\tau _{{\rm{main}}}}$为自相关函数的第1零点，${\tau _{3{\text{ dB}}}}$为自相关函数的3 dB点 旁瓣分辨性能 积分旁瓣能量：${{\rm{ISE}}} = \dfrac{1}{ {|\chi (0){|^2} } }\left[ {\displaystyle\int_{ - T}^{ - {\tau _{ {\text{main} } } }} {|\chi (\tau ){|^2}{\text{d} }\tau } + \displaystyle\int_{ {\tau _{ {\text{main} } } }}^T {|\chi (\tau ){|^2}{\text{d} }\tau } } \right]$
峰值旁瓣：$\mathrm{max}\left(|\chi (\tau ){|}^{2}\right),\tau \in \left[-T, -{\tau }_{\text{main} }\right]\cup \left[{\tau }_{\text{main} },T\right]$，加权积分旁瓣：${{\rm{WeIC}}}{ {{\rm{E}}}_\gamma } = \gamma \cdot {{\rm{IME}}} + {{\rm{ISE}}}$，$ \chi (\tau ) $为信号自相关函数
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表 5 常见信号约束条件
Table 5. Common signal performance evaluation criteria
约束条件 表达形式 能量约束 ${{\boldsymbol{s}}^{\text{H}}}{\boldsymbol{s}} \le E$ 峰均比约束 ${{\rm{PAPR}}} ({\boldsymbol{s} }) = \dfrac{ {\mathop {\max }\limits_k { {\left| { {s_k} } \right|}^2} } }{ {\dfrac{1}{N}\displaystyle\sum\limits_{k = 1}^N { { {\left| { {s_k} } \right|}^2} } } } = \dfrac{ {\mathop {\max }\limits_k { {\left| { {s_k} } \right|}^2} } }{ {E/N} } \le \mu$ 恒包络约束 $\Vert {\boldsymbol{s}}\Vert =1$ 相似性约束 ${\left\| {{\boldsymbol{s}} - {s_0} } \right\|^2} \le \varepsilon$
$ \varepsilon $是用来约束相似性程度，$ {s_0} $为参考的基准信号
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表 6 干扰功率谱描述参数
Table 6. Interference power spectrum description parameters
参数 含义 干扰频带
中心频率$\left[ { {f_1},{f_2}{\text{,} }\cdots{\text{,} }\,{f_N} } \right]$
矩形扩展模型阻带中心频率干扰源带宽 $\left[ { {B_{\varOmega 1} },{B_{\varOmega 2} }{\text{,} }\cdots{\text{,} }\,{B_{\varOmega N} } } \right]$, 每个干扰源占据的带宽 完整可用频带 $ \tilde B $ 可用比 $\rho = { {\left( {\tilde B - \displaystyle\sum\nolimits_{i = 1}^N { {B_{\varOmega i} } } } \right)}/ {\tilde B} }$ 非连续比 $p = \left( {\displaystyle\sum\nolimits_{i = 1}^N { {B_{\varOmega i} } } } \right)/\tilde B$
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表 7 优化问题形式
Table 7. Optimization problem form
问题形式 表达式 二次型优化问题 $\begin{gathered} \mathop { {\text{min} } }\limits_{\boldsymbol{s} } \displaystyle\sum\limits_{k = 1}^M { {\gamma _k}{ {\boldsymbol{s} }^{\text{H} } }{ {\boldsymbol{R} }_k}{\boldsymbol{s} } } {\text{ + } }\displaystyle\sum\limits_{l = 1}^N { {\gamma _l}\left| { { {\boldsymbol{s} }^{\text{H} } }{ {\boldsymbol{R} }_l}{\boldsymbol{s} } } \right|} \hfill \\ {\text{s} }{\text{.t} }{\text{. PAR} }\left( {\boldsymbol{s} } \right) \le \alpha /{\text{PAR} }\left( {\boldsymbol{s} } \right) = 1 \hfill \\ {\text{ } }{ {\text{l} }_\varOmega } \le \left| { { {\text{a} }^{\text{H} }_\varOmega }{\boldsymbol{s} } } \right| \le { {\text{u} }_\varOmega } \hfill \\ {\text{ } }\left| {{\boldsymbol{s}} - {{\boldsymbol{s}}_r} } \right| \le \varepsilon \hfill \\ \end{gathered}$ 四次型优化问题 $\begin{array}{l}\underset{ {\boldsymbol{s} } }{\mathrm{min} }\displaystyle\sum _{n=1}^{N}{\gamma }_{n}{\left({ {\boldsymbol{s} } }^{\text{H} }{\boldsymbol{R}}{\boldsymbol{s} }-{p}_{n}\right)}^{2}\\ \text{s}\text{.t}\text{. PAR}\left({\boldsymbol{s} }\right)\le \alpha /\text{PAR}\left({\boldsymbol{s} }\right)=1\\ {\text{ l} }_{\varOmega }\le \left|{\text{a} }_{\varOmega }^{\text{H} }{\boldsymbol{s} }\right|\le {\text{u} }_{\varOmega }\end{array}$ 极大极小型优化问题 $\begin{array}{l}\underset{ {\boldsymbol{s} } }{\text{min } }\text{max}\; {f} \left({\boldsymbol{s}}\right)\\ \text{s}\text{.t}\text{. PAR}\left({\boldsymbol{s} }\right)\le \alpha /\text{PAR}\left({\boldsymbol{s} }\right)=1\\ {\text{ l} }_{\varOmega }\le \left|{\text{a} }_{\varOmega }^{\text{H} }{\boldsymbol{s} }\right|\le {\text{u} }_{\varOmega }\\ \text{ }\left|{\boldsymbol{s} }-{ {\boldsymbol{s} } }_{r}\right|\le \varepsilon \end{array}$ 注：s为待合成信号，R为性能矩阵($ {{\boldsymbol{R}}_k} $为频域抗干扰性能矩阵、$ {{\boldsymbol{R}}_l} $为分辨性能矩阵)，${f} \left( {\boldsymbol{s}} \right)$为信号性能函数，$ {p_n} $为频谱模板，$ \gamma $为准则加权系数，${\text{PAR} }\left( {\boldsymbol{s}} \right)$为信号的峰均功率水平($ 0 \le \alpha \le 1 $)，当且仅当$ \alpha {\text{ = }}1 $时信号恒模，$\left| { {\text{a} }_\varOmega^{\text{H} }{\boldsymbol{s}}} \right|$是信号的离散频谱，$ {{\text{u}}_\varOmega } $和$ {{\text{l}}_\varOmega } $分别为频谱约束上、下限，$ \varOmega $为受限频率的集合，$\left| {{\boldsymbol{s}} - {{\boldsymbol{s}}_r} } \right| \le \varepsilon$为相似性约束，${{\boldsymbol{s}}_r}$为某些具有探测性能的信号(如LFM信号)。
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